Binomial Distribution Calculator
Compute exact and cumulative binomial probabilities for n independent trials with success probability p, including mean, variance, standard deviation, mode and a step-by-step breakdown of the formula.
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isCalculator Editorial Team
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Frequently Asked Questions
The probability of exactly k successes in n independent trials with success probability p is P(X = k) = C(n, k) · p^k · (1 − p)^(n − k), where C(n, k) = n! / (k! · (n − k)!) is the binomial coefficient counting the number of ways to choose which k trials succeed.
n is the total number of independent trials, p is the probability of success on a single trial (entered as a decimal between 0 and 1, so 0.5 means 50%), and k is the number of successes you want to evaluate. The calculator then returns the exact probability, cumulative probabilities, and the distribution's mean, variance and standard deviation.
P(X = k) is the probability of exactly k successes. P(X ≤ k) is the cumulative probability of k or fewer successes (the lower tail), while P(X ≥ k) is the probability of k or more successes (the upper tail). The calculator lists all related probabilities so you can compare them at a glance.
The mean (expected number of successes) is n·p, the variance is n·p·(1 − p), and the standard deviation is the square root of the variance. The most likely number of successes (the mode) is the floor of (n + 1)·p.
Yes. When p = 0 the distribution is degenerate: P(X = 0) = 1 and every other probability is 0. When p = 1, P(X = n) = 1 instead. The calculator detects these cases automatically and explains them in the step-by-step breakdown.
